1,249 research outputs found
Affordable, Entropy Conserving and Entropy Stable Flux Functions for the Ideal MHD Equations
In this work, we design an entropy stable, finite volume approximation for
the ideal magnetohydrodynamics (MHD) equations. The method is novel as we
design an affordable analytical expression of the numerical interface flux
function that discretely preserves the entropy of the system. To guarantee the
discrete conservation of entropy requires the addition of a particular source
term to the ideal MHD system. Exact entropy conserving schemes cannot dissipate
energy at shocks, thus to compute accurate solutions to problems that may
develop shocks, we determine a dissipation term to guarantee entropy stability
for the numerical scheme. Numerical tests are performed to demonstrate the
theoretical findings of entropy conservation and robustness.Comment: arXiv admin note: substantial text overlap with arXiv:1509.06902;
text overlap with arXiv:1007.2606 by other author
A Topical and Narrative Analysis of Napoleonic Era Battle Pieces
The instrumental battle piece, as it was at the time of the Napoleonic Wars, was a genre largely reserved for amateur musicians. However, within these commercial compositions there was a distinct topical language. Analysis of multiple pieces that discuss the same battle, but come from composers of different nations reveals that composers of different nationalities often changed the narrative of the battle being portrayed in order to paint their country—or adopted country— in a better light. This can be seen in a comparison of compositions based on the battles of Trafalgar and Waterloo, by composers writing for English and Germanic audiences. This was likely due to composers pandering to their intended audience; a composition intended for publication in England is going to see increased sales if the English forces play a prominent and positive role in the narrative. Daniel Steibelt is the exception to these general traits; His battle pieces are written for his own use as a concert pianist, not the commercial music market. As such, the topical language employed by Steibelt in his compositions is on a whole more creative than other composers, and elements of theater are often incorporated, as are popular songs and opera arias. In addition, Steibelt traveled regularly during the time period, and thus his output includes compositions written from both the pro-French and anti-French perspectives. Though the genre of the battle piece is often critically panned, examples of the topical techniques developed in these pieces can be found in more respected orchestral works that depict battle from the mid to late Romantic era, including works by Tchaikovsky and Liszt. Thus, musicologists and others who write off the compositions of Steibelt and his contemporaries are ignoring a part of the repertoire that clearly influenced these later canonical works
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis
of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD
equations as they are particularly useful for mathematically modeling a wide
variety of magnetized fluids. In order to be self-contained we first motivate
the physical properties of a magnetic fluid and how it should behave under the
laws of thermodynamics. Next, we introduce a mathematical model built from
hyperbolic partial differential equations (PDEs) that translate physical laws
into mathematical equations. After an overview of the continuous analysis, we
thoroughly describe the derivation of a numerical approximation of the ideal
MHD system that remains consistent to the continuous thermodynamic principles.
The derivation of the method and the theorems contained within serve as the
bulk of the review article. We demonstrate that the derived numerical
approximation retains the correct entropic properties of the continuous model
and show its applicability to a variety of standard numerical test cases for
MHD schemes. We close with our conclusions and a brief discussion on future
work in the area of entropy consistent numerical methods and the modeling of
plasmas
An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
We extend the entropy stable high order nodal discontinuous Galerkin spectral
element approximation for the non-linear two dimensional shallow water
equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J.
Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin
method for the two dimensional shallow water equations on unstructured
curvilinear meshes with discontinuous bathymetry. Journal of Computational
Physics, 340:200-242, 2017] with a shock capturing technique and a positivity
preservation capability to handle dry areas. The scheme preserves the entropy
inequality, is well-balanced and works on unstructured, possibly curved,
quadrilateral meshes. For the shock capturing, we introduce an artificial
viscosity to the equations and prove that the numerical scheme remains entropy
stable. We add a positivity preserving limiter to guarantee non-negative water
heights as long as the mean water height is non-negative. We prove that
non-negative mean water heights are guaranteed under a certain additional time
step restriction for the entropy stable numerical interface flux. We implement
the method on GPU architectures using the abstract language OCCA, a unified
approach to multi-threading languages. We show that the entropy stable scheme
is well suited to GPUs as the necessary extra calculations do not negatively
impact the runtime up to reasonably high polynomial degrees (around ). We
provide numerical examples that challenge the shock capturing and positivity
properties of our scheme to verify our theoretical findings
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
We design an arbitrary high-order accurate nodal discontinuous Galerkin
spectral element approximation for the nonlinear two dimensional shallow water
equations with non-constant, possibly discontinuous, bathymetry on
unstructured, possibly curved, quadrilateral meshes. The scheme is derived from
an equivalent flux differencing formulation of the split form of the equations.
We prove that this discretisation exactly preserves the local mass and
momentum. Furthermore, combined with a special numerical interface flux
function, the method exactly preserves the mathematical entropy, which is the
total energy for the shallow water equations. By adding a specific form of
interface dissipation to the baseline entropy conserving scheme we create a
provably entropy stable scheme. That is, the numerical scheme discretely
satisfies the second law of thermodynamics. Finally, with a particular
discretisation of the bathymetry source term we prove that the numerical
approximation is well-balanced. We provide numerical examples that verify the
theoretical findings and furthermore provide an application of the scheme for a
partial break of a curved dam test problem
A Provably Stable Discontinuous Galerkin Spectral Element Approximation for Moving Hexahedral Meshes
We design a novel provably stable discontinuous Galerkin spectral element
(DGSEM) approximation to solve systems of conservation laws on moving domains.
To incorporate the motion of the domain, we use an arbitrary
Lagrangian-Eulerian formulation to map the governing equations to a fixed
reference domain. The approximation is made stable by a discretization of a
skew-symmetric formulation of the problem. We prove that the discrete
approximation is stable, conservative and, for constant coefficient problems,
maintains the free-stream preservation property. We also provide details on how
to add the new skew-symmetric ALE approximation to an existing discontinuous
Galerkin spectral element code. Lastly, we provide numerical support of the
theoretical results
Regularized lattice theory for spatially dispersive nonlinear optical conductivities
Nonlinear optical responses are becoming increasingly relevant for
characterizing the symmetries and quantum geometry of electronic phases in
materials. Here, we develop an expanded diagrammatic scheme for calculating
spatially dispersive corrections to nonlinear optical conductivities, which we
expect to enhance or even dominate even-order responses in materials of recent
interest. Building upon previous work that enforces gauge invariance of
spatially uniform nonlinear optical responses, we review the cancellation of
diagrams required to ensure the equivalence between velocity gauge and length
gauge formulations, and provide a simple vertex rule for extending optical
responses to first order in the light wave vector q. We then demonstrate the
method with calculations on a prototypical centrosymmetric model where spatial
dispersion admits anomalous secondharmonic generation, a response that is
symmetry-forbidden under the dipole approximation.Comment: 14 pages, 9 figure
Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes
Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e. the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. The source code to reproduce all numerical experiments presented in this article is available online (DOI: 10.5281/zenodo.4054366)
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